Compositions of theta correspondences (Q703805)

This paper is concerned with the interplay, by chains of theta correspondences, between the representations of groups of the form \(O(p,q)\) (\(p+q \equiv 0 \pmod 2\)) and \(Sp(2n)\) over \(\mathbb R\). If one begins with a representation \(\pi\) of a group \(O(p_1,q_1)\), then uses the theta correspondence to get one of \(Sp(2n_1)\), then again one of \(O(p_2,q_2)\) and so on one has a process the author calls ``quantum induction''. The question investigated here is whether, if the original representation \(\pi\) is unitary, the result of this process is also unitary. The author answers this in a number of ranges when just two steps are applied; some of the results are of the form that the representation is unitary or zero. The author also gives a conjectural description of the image in terms of parabolic induction. The methods used in the proofs are those developed by the author and J.-S. Li and others recently in the theory of the theta correspondence.